UKMT Senior Challenge-IV

We want the solutions to $x^2+y^2+z^2 = 3^2=9$ . We see that $9$ as a sum of 3 squares is either $1+4+4$ or $0+0+9$ , giving the solutions $(0,0, \pm 3), (\pm 1, \pm 2, \pm 2)$ and permutations. Counting them all, we have $\boxed{30}$ solutions.

Easiest way to visualize this is to see that there are 3 such points in each of the 8 quadrants, plus the 6 points on the axes.